1 Introduction

Ultraluminous X-ray sources (ULXs) are off-nuclear point-like sources that have intrinsic X-ray luminosities \({\ge} 10^{39}~\mbox{ergs}\,\mbox{s}^{-1}\) in the 0.5–10.0 keV energy range and up to \({\sim} 10^{41\mbox{--}42}~\mbox{ergs}\,\mbox{s}^{-1}\) or so to say above the Eddington limit of a \(20~M_{\odot}\) black hole (Feng and Soria 2011). Einstein was the first X-ray satellite to reveal the existence of X-ray sources in the external galaxies (Fabbiano 1989). However recent observations with much improved observatories, such as, Chandra and XMM-Newton, have greatly contributed in the increased detections of this kind of sources (Feng and Soria 2011). These high luminosities are explained as due to super-Eddington accretion on to stellar mass BHs or neutron stars (Begelman 2002) and sub-Eddington accretion on to Intermediate Mass Black Holes (IMBHs) (Colbert and Mushotzky 1999). Although the true nature of these objects is under study, current models propose several alternatives to explain their high luminosities. It could be either due to the geometric beaming if ULXs are powered by an accreting stellar black hole (King et al. 2001) or the super-Eddington fluxes originating from disks (Begelman 2002). These high luminosities imply black holes (BHs) of masses greater than a \(10~M_{\odot}\), if the accretion is sub-Eddington and the radiation is isotropic. This suggested that ULXs could harbor Intermediate Mass Black Holes (IMBHs) i.e. Black Hole masses in the range \(10~M_{\odot} < M < 10^{5}~M_{\odot}\) (Colbert and Mushotzky 1999). The presence of a soft (0.1–0.2 keV) component also agrees with the IMBH theory (Kaaret et al. 2003). Earnshaw et al. (2016) has reported that a ULX (ULX-7) in the galaxy M51 to be an IMBH accreting in the hard state with photon index around 1.5 and a black hole mass upper limit of \({\sim} 3.5\times 10^{4}~M_{\odot}\). The alternative is stellar-remnant Black Holes accreting at around or above the Eddington limit (e.g., King et al. 2001; Feng and Soria 2011). It has been proposed that ULXs may only seem to exceed the Eddington Limit due to one or other reasons. There are many ways an accreting source can appear to exceed the Eddington Limit, like relativistic beaming (Freeland et al. 2006) or geometrical beaming (Paczynsky and Wiita 1980), which make these sources appear to emit such high intensities. A viable explanation of the high ULX luminosity can be given by geometrical beaming in combination with super-Eddington accretion where the emission is non-isotropic (Abramowicz et al. 1980). Ultraluminous X-ray sources with luminosities \({>} 10^{41}~\mbox{ergs}\,\mbox{s}^{-1} \) located away from the nucleus of external galaxies are being termed as Hyperluminous X-ray sources, HLXs (Gao et al. 2003). In order to produce such extreme luminosities with sub-Eddington accretion, these HLXs are required to host Intermediate Mass Black Holes with masses \({>} 10^{3}~M_{\odot}\) (Sutton et al. 2012). Farrell et al. (2009) reported the discovery of a variable X-ray source with maximum 0.2–10 keV luminosity of \({\sim} 1.1 \times 10^{42}~\mbox{ergs}\,\mbox{s}^{-1}\) in the spiral galaxy ESO 243-49. The mass of the Black hole associated was estimated to be around 500 \(M_{\odot}\). Another ULX (X-1), in the nearby galaxy M82, with an X-ray luminosity of \(10^{41}~\mbox{ergs}\,\mbox{s}^{-1}\) (Qiu et al. 2015) is considered to be one of the best IMBH candidate, (Pasham et al. 2014) estimated its mass to be \(438 \pm 105~M_{\odot}\). Other point sources like Cartwheel N10 (Gao et al. 2003), 2XMM J011942.7 + 032421 in NGC 470 (Walton et al. 2011) are considered to belong to the Hyperluminous category.

However many lower luminosity ULXs (\(L < 10^{41}~\mbox{ergs}\,\mbox{s}^{-1}\)) are assumed to be accreting above the Eddington Limit because their high luminosities are not due to beaming as most of the ULXs are seen to change state. It is seen in M31 that two X-ray binaries changed into ultraluminous state and then returned to their regular states (Middleton et al. 2013). These transitions showed that ULXs can be an ultraluminous state of accreting compact objects. It can also be said that ULXs are higher stellar mass black holes (\(20~M_{\odot} < M_{\mathit{BH}} <100~M_{\odot} \)) accreting below the Eddington limit (Mapelli and Zampieri 2014; Zampeiri and Roberts 2009). Devi and Singh (2014) reported the discovery of an ULX (X-8) in the galaxy NGC 3384 having a 0.3–10 keV luminosity of \(1.52 \times 10^{39}~\mbox{ergs}\,\mbox{s}^{-1}\) and a bolometric luminosity of \(2.43 \times 10^{39}~\mbox{ergs}\,\mbox{s}^{-1}\). ULX X-8 had hard X-ray spectra with photon index around 1.97 and appeared to be consistent with nearly a \(30~M_{\odot} \) black hole accreting at around 0.58 times its Eddington luminosity. Some models suggest that ULXs are a bright, short-lived, but common phase of stellar mass X-ray binary evolution as observations support that apparently super-Eddington episodes are common in stellar mass binaries (King et al. 2001). Also many other models have been proposed regarding ULXs, showing some ULXs to be supernova remnants (Mezcua and Lobanov 2011), accreting white dwarfs (Xu et al. 2012) and background AGNs (Dadina et al. 2013). Sutton et al. (2015) reported the HLX 2XMM J134404.1-271410 in IC 4320 to be a background AGN.

2 Observation and data analysis

Chandra Advanced CCD Imaging Spectrometer (ACIS) has observed NGC 5643 three times and NGC 7457 four times. The observation summary is listed in Table 1. In this work we have selected the data from 2004 (Obs ID 5636) and the data with higher exposure from 2015 (Obs Id 17031) for NGC 5643 and data from 2003 (Obs ID 4697), 2009 (Obs ID 11786)and 2015 (Obs ID 18440, having higher exposure) for NGC 7457. The data reduction and analysis were done using Chandra Interactive Analysis of Observations (CIAO) version-4.8 and Heasoft-6.18. The CALDB 4.7.0 distributed with CIAO-4.8 is used. The X-ray point sources were extracted from the level 2 event lists by using the CIAO source detection tool WAVDETECT. The sources having data counts above 100 were selected for the spectral analysis so that the spectral parameters could be constrained properly using two-parameter model. The background regions were selected as a source free region near the corresponding sources. Sources near the nucleus, where nearby source-free background regions are difficult to be found, are excluded. Also sources having count rate \({>} 0.05~\mbox{counts}\,\mbox{s}^{-1}\) are excluded to avoid photon pileup effects. A combination of CIAO tools and calibration data is used to extract the source and background spectra. The spectra are then grouped and binned using grppha from Heasoft. Using spectral fitting package XSPEC version 12.9.0, available in the Heasoft package, the spectral analysis was done and the data were fitted in the energy range 0.3–10 keV. The sources were fitted using two spectral models viz. the absorbed powerlaw and an absorbed disk blackbody, absorption was taken into account by using the XSPEC model phabs. The C-statistics was used for the analysis. Devi et al. (2007) analysed 365 sources from thirty galaxies available at the Chandra Archive, using two models- an absorbed powerlaw and an absorbed disk blackbody model, and presented the dependence of luminosities of X-ray point sources on spectral models. A better fit to the data was determined either by powerlaw model or disk blackbody model when the C-statistic difference between the models was \({\ge} 2.7\). Same method was adopted by Devi and Singh (2014) while studying the Chandra observed X-ray point sources of NGC 3384. However, for those sources which can be explained well by both the models, the lower of the estimated luminosities from the models was chosen by them. Same method has also been adopted for the present work.

Table 1 Observation log of NGC 5643 and NGC 7457
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The black hole mass harboured by ULXs can be estimated using the disk blackbody model (Devi and Singh 2014), by assuming the inner disk radius, \(R_{\mathit{in}} \) to be \({\sim} 10~\mbox{GM}/\mbox{c}^{2} \) (taking the values of viewing angle, cosi = 0.5 and color factor, f = 1.7). The black hole masses can also be estimated using the powerlaw model. Zhou and Zhao (2010) presented a correlation between powerlaw photon index (\(\varGamma\)) and both Eddington ratio and bolometric correction of reverberation mapped AGNs as:

\(\log{(L_{\mathit{bol}}/L_{\mathit{Edd}})}= (2.09 \pm 0.68) \varGamma - (4.98 \pm 1.04)\)

\(\log{(L_{\mathit{bol}}/L_{2\mbox{--}10~\mbox{keV}})}= (1.12 \pm 0.30) \varGamma - (0.63 \pm 0.53)\)

If we assume that ULXs follow the same correlations, the \(M_{\mathit{BH}}\) \((L_{\mathit{Edd}})\) can be estimated from the photon index \(\varGamma\) and \(L_{2\mbox{--}10\,\text{keV}} \) obtained from the powerlaw model. Here in this work, we have estimated the masses of the black holes by both these methods and compared the result.

Using dmextract tool of CIAO, the lightcurve of all the sources were also extracted to check for any signature of the sources varying in kiloseconds time-scale.

Table 2 lists the celestial positions and spectral properties of the X-ray point sources.

Table 2 Spectral Properties of the Point Sources
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3 Results and discussions

3.1 NGC 5643

NGC 5643 is a nearby Seyfert 2 galaxy. Excluding the diffuse nuclear region, a total of four X-ray point sources (X-1, X-2, X-3 and X-4) from Obs ID 17031 and one source (equivalent to X-4 of Obs ID 17031) from Obs ID 5636 having counts greater than 100 are obtained. The source obtained in Obs ID 5636 is found to have a count rate \({>} 0.05 \) counts \(\mbox{s}^{-1}\), and hence excluded as it will be affected by photon pileup. We report the three sources X-1, X-2 & X-3 to be ULX candidates, which can be explained by both the models- an absorbed powerlaw and an absorbed diskblackbody model. The luminosity of X-1, X-2 & X-3 as calculated by both the above mentioned models is found to be of the order of \(10^{39}~\mbox{ergs}\,\mbox{s}^{-1} \) while X-4 to be around \(10^{40}~\mbox{ergs}\,\mbox{s}^{-1} \). Spectral analysis reveals these four sources are in hard state with the powerlaw photon index (\(\varGamma \)) in the range \(1.42 \mbox{--} 1.86\). This is indicative that the radiative mechanism of these hard ULX sources is probably inverse comptonization of soft photons. Also their spectra, when fitted by disk blackbody model, is found to be relatively hard with their inner disk temperature (kT) to be in the range \((1.08 \mbox{--} 1.80)~\mbox{keV} \). The spectra of these sources are shown in Fig. 1 and Fig. 2. The disk blackbody model estimates these four sources to harbour stellar mass BHs: \(\mbox{X-1} \sim 9.07^{+8.64}_{-5.52}~M_{\odot} \), \(\mbox{X-2} \sim 11.91^{+37.95}_{-9.26}~M_{\odot} \), \(\mbox{X-3} \sim 20.96^{+12.53}_{-6.66}~M_{\odot} \) and \(\mbox{X-4} \sim 31.15^{+2.54}_{-2.36}~M_{\odot} \). For those sources having low counts (such as X-1, X-2), the black hole mass could not be constrained properly. However, all of them are estimated to be stellar mass black holes with upper limit \({<} 50~M_{\odot}\). It is observed that the sources X-1 and X-3 are accreting at a sub-Eddington rate, whereas the sources X-2 and X-4 at super-Eddington rate. The result for X-4 is consistent with the results of an independent analysis using XMM-Newton data (Pintore et al. 2016). While, if ULXs are assumed to follow the same correlations of Zhou and Zhao (2010) the black hole masses estimated from the powerlaw model for all the four sources are found to be of the order of \({\sim} 10^{3\mbox{--}4}~M_{\odot}\) which differs significantly from the mass estimated using disk blackbody model. This may be indicative that ULXs may not be accreting at a rate similar to that of AGNs. X-4 was first reported by Guainazzi et al. (2004) (christened NGC 5643 X-1) to have a luminosity of \(4.4 \times 10^{40}~\mbox{ergs}\,\mbox{s}^{-1}\), based on the 2003 XMM-Newton data. Krivonos and Sazonov (2016) studied the NuSTAR and XMM-Newton data and reported the source to brighten from luminosity \({\sim} 1.5 \times 10^{40}~\mbox{ergs}\,\mbox{s}^{-1}\) in 2009 to \(\sim 3 \times 10^{40}~\mbox{ergs}\,\mbox{s}^{-1}\) in 2014. Here in the present work using Chandra data (2015), X-4 is found to have a luminosity of \({\sim} 2.27 \times 10^{40}~\mbox{ergs}\,\mbox{s}^{-1}\). Thus, if we take into consideration all these luminosity fluctuations of source X-4 from XMM, NuSTAR and Chandra data, this source can be considered as a ULX source varying its luminosity in years.

Fig. 1
figure 1

The absorbed disk blackbody spectrum of the X-Ray Sources in NGC 5643

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Fig. 2
figure 2

The absorbed powerlaw spectrum of the X-Ray Sources in NGC 5643

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Temporal analysis of these sources has been carried out to check for variability in kilo-seconds time-scale. The light curves of these sources, binned over 0.5, 1, 2 and 4 ks are shown in Fig. 3. The temporal study shows that the variability in kilo-seconds time-scales in all these sources are absent as the probability that the count rate was constant during the observation is all \({>} 0.2 \).

Fig. 3
figure 3

The Light curves of the X-ray point sources in NGC 5643

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3.2 NGC7457

NGC 7457 is a nearby early type galaxy. Here, for Obs ID 4697, no point sources having counts above 100 was detected. In Obs ID 11786 one point source (X-5) and from Obs Id 18440 two point sources (one similar to X-5 of ObsID 11786 and another X-6) having counts above 100 are detected. Spectral analysis of X-5 reveals its luminosity to be \(1.4 \times 10^{39}~\mbox{ergs}\,\mbox{s}^{-1}\) in the energy range (0.3–10) keV with a hard Powerlaw Photon Index, \(\varGamma \sim 1.8 \) which may be due to inverse comptonization of soft photons. If the spectrum is explained by disk blackbody model, X-5 is found to have a bolometric luminosity \(2.0 \times 10^{39}~\mbox{ergs}\,\mbox{s}^{-1}\) with an inner disk temperature, \(\mbox{kT} \sim 1.11\) keV with ObsID 11786. Almost similar result is found with that of ObsID 18440 with a luminosity of \(1.4 \times 10^{39}~\mbox{ergs}\,\mbox{s}^{-1}\) in the energy range (0.3–10) keV with a hard Powerlaw Photon Index, \(\varGamma \sim 1.86 \) and a bolometric luminosity \(1.9 \times 10^{39}~\mbox{ergs}\,\mbox{s}^{-1}\) with an inner disk temperature, \(\mbox{kT} \sim 1.48\) keV.

The spectra of this source is shown in Fig. 4 and Fig. 5. The disk blackbody model estimates that this ULX harbours a stellar mass black hole \({\sim} 10.81^{+5.82}_{-4.12}~M_{\odot} \) accreting at \({\sim} 1.48\) times the Eddington luminosity. Also the black hole mass estimated from the powerlaw model (Zhou and Zhao 2010) is found to be of the order of \(10^{3}~M_{\odot}\) which is in contrast to the result obtained from the disk blackbody model, as found in above cases. Hence, it may be true that ULXs may not be accreting at the same rate as AGNs.

Fig. 4
figure 4

The absorbed disk blackbody spectrum of X-5 & X-6

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Fig. 5
figure 5

The absorbed powerlaw spectrum of X-5 & X-6

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To check the presence of kiloseconds variability, temporal analysis of this source has been carried out. Figure 6 shows the light curve of this source binned over 0.5, 1, 2 and 4 ks. The probability that the source is not variable is all these time bins \({\ge} 0.5 \). Hence signature for this source to be variable in kilo-seconds time-scale is absent.

Fig. 6
figure 6

The Light curves of the X-5

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X-6 is a nuclear source which is well aligned with the optical centre of the galaxy NGC 7457. The spectrum of this source is marginally better fitted by powerlaw model with luminosity \({\sim} 7.07 \times 10^{38}~\mbox{ergs}\,\mbox{s}^{-1}\) in the (0.3–10) keV energy range and with a relatively soft photon index (\(\varGamma \sim 2.74 \)) which is consistent with the result of Peacock et al. (2017).

4 Summary

Chandra observations of NGC 5643 and NGC 7457 has been analyzed. A total of 4 sources were found in NGC 5643 and 2 sources in NGC 7457 having counts \({\ge} 100\). Using a combination of CIAO tools and calibration data, the source (and background) spectrum are extracted. Spectral analysis was performed using XSPEC version 12.9.0. The spectra of the point sources were fitted using the two models:absorbed powerlaw and absorbed disk blackbody. Three Ultraluminous X-ray sources, ULXs (X-1, X-2 and X-3) were detected in NGC 5643. Both the models estimate the luminosities of these 3 sources to be of the order of \(10^{39}\) with X-1 having \(\varGamma = 1.42\) and \(kT = 1.52\) keV, X-2 having \(\varGamma = 1.75\) and \(kT = 1.35\) keV and X-3 having \(\varGamma = 1.86 \) and \(kT = 1.08 \) keV. The spectral analysis of these sources suggest that all the three ULXs are in hard states and harbouring stellar mass black holes with X-1 & X-3 accreting at sub-Eddington rate while X-2 at a super-Eddington rate. X-4 is a ULX varying its luminosity in years and harbouring a stellar mass BH of \({\sim} 30~M_{\odot}\) accreting at a super-Eddington rate. Another ULX (X-5) in NGC 7457 is detected with luminosity \(1.4 \times 10^{39}~\mbox{ergs}\,\mbox{s}^{-1}\) in the energy range (0.3–10) keV with a hard powerlaw Photon Index, \(\varGamma \sim 1.86 \) and a bolometric luminosity \(1.9 \times 10^{39}~\mbox{ergs}\,\mbox{s}^{-1}\) with an inner disk temperature, \(\mbox{kT} \sim 1.48~\mbox{keV}\). The mass of the black hole harboured is estimated to be \({\sim} 10.81^{+5.82}_{-4.12}~M_{\odot} \) accreting at a high rate of \({\sim}1.48\) times its Eddington luminosity. The light curve of these sources binned at different (ks) time bins have also shown the absence of short term variability in kilo-seconds time scales.